Table Of ContentEcho Tomography of Black Hole Accretion Flows a
Keith Horne b
University of St.Andrews, KY16 9SS, Scotland, UK
ABSTRACT
3
We discuss technologies for micro-arcsec echo mapping of black hole accretion flows in Active Galactic Nuclei
0
(AGN). Echo mapping employs time delays, Doppler shifts, and photoionisation physics to map the geometry,
0
kinematics,andphysicalconditionsinthereprocessingregionclosetoacompacttime-variablesourceofionizing
2
radiation. Timedelaymapsarederivedfromdetailedanalysisofvariationsinlightcurvesatdifferentwavelengths.
n
Echo mapping is a maturing technology at a stage of development similar to that of radio interferometry just
a
J before the VLA. The first important results are in, confirming the basic assumptions of the method, measuring
4 the sizes of AGN emission line regions, delivering dozens of black hole masses, and showing the promise of the
1 technique. Resolution limits with existing AGN monitoring datasets are typically 5-10light days. This should
improve down to 1-2 light days in the next-generation echo mapping experiment∼s, using facilities like Kronos
1
andRobonetthataredesignedforanddedicatedtosustainedspectroscopicmonitoring. Alightdayis0.4micro-
v
arcsecataredshiftof0.1,thusechomappingprobesregions103 timessmallerthanVLBI,and105 timessmaller
0
5 than HST.
2
1
1. MICRO-ARCSEC ECHO TOMOGRAPHY
0
3 Angularresolutionisadiscoveryfrontierthatdrivesthedevelopmentofastronomicaltechnology. HSTdelighted
0
the public andtransformedastronomybydeliveringstunning sub-arcsecimagesatultraviolet,optical,andmore
/
h recently infrared wavelengths. Large ground-based telescopes can now produce sub-arcsec images at infrared
p wavelengths by using adaptive optics to compensate for atmospheric turbulence. Sub-arcsec radio maps are
-
o routinely constructed from complex visibility measurements obtained with multi-element radio interferometers
r (VLA,Merlin),andmilli-arcsecresolutionisachievedwiththeVeryLongBaselineArray(VLBA).Themostam-
t
s bitioustechnologyonthe angularresolutionhorizonistheMicro-ArcsecX-rayInterferometryMission,MAXIM,
a
which may one day image the flow of material into nearby black hole event horizons.
:
v
Today we are alreadyexploring the micro-arcsecstructure of black hole accretionflows through echo tomog-
i
X raphy experiments. Tomography is an indirect imaging technique that recovers an image from measurements of
r projectionsofthatimage. Tomographyisanalogoustointerferometry,exceptthatthe measurementsareofpro-
a
jections rather than fourier components of the image. Echo mapping employs time-resolved spectrophotometry
to record spectral variations in which detailed information on spatial and kinematic structure is coded as time
delays and Doppler shifts. This article discusses echo tomography technologies in use and under development
for micro-arcsec mapping of black hole accretion flows.
1.1. Black Hole Accretion Flows
Black holes come in two varieties: stellar-mass holes (M 10M ) in X-ray binaries, and supermassive black
⊙
∼
holes (M 106−9M ) in the nuclei of galaxies. Accretion disks convey angular momentum outward as matter
⊙
∼
spirals inward to feed the central black hole. Accretion theory and black hole scaling laws can be tested by
comparingphenomenainX-raybinariesandactivegalacticnuclei(AGN).Althoughtheyaretoosmallfordirect
or interferometric imaging, these accretion flows can be resolved by echo tomography.
Current models of AGN spectral energy distributions include optical to X-ray emission from the accretion
disk,infraredemissionfromadustymoleculartorusencirclingandfromsomeviewinganglesobscuringthe disk,
and relativistically beamed radio to X-ray emission from jets. Emission lines arise from reprocessing of X-ray
aToappearin: SPIE.AstronomicalTelescopesandInstrumentation. FutureEUV-UVandVisibleSpaceAstrophysics
Missions and Instrumentation. Waikoloa HI. 22-28 Aug 2002. Eds. J.C.Blades, O.H.Siegmund. SPIEProc 4856.
b Email: [email protected],Web: http://star-www.st-and.ac.uk/∼kdh1
Figure 1. Ionizing photons from a compact source are reprocessed by gas clouds in a thin spherical shell. A distant
observer sees the reprocessed photons arrive with a time delay ranging from 0 for reprocessing at the near edge of the
shellto2R/cforreprocessing atthefaredge. Theiso-delay paraboloids slice theshellintozoneswith areas proportional
to therange of delays.
and EUV radiation in two kinematically-distinct regions, the broad line region (BLR) with V 104 km s−1,
andthe narrow-lineregion(NLR)withV <103 km s−1. TheNLR,resolvedbyHST,hasabi-pol∼armorphology
interpreted as wide cones of ionising radiation emerging from the unresolved nucleus. The accretion disk and
BLR are unresolved, but these regions vary on timescales of days to months in response to erratic changes
in ionising radiation from regions still closer to the black hole. Light travel time within the system produces
observable time delays, providing indirect information on the size and structure of the disk and BLR, which
make these regions accessible to echo tomography.
1.2. Time Delay Paraboloids
Reverberationmapping1 reliesonacompacterraticallyvariablesourceofionizingradiationembeddedwithinthe
region that we wish to probe. Heated and photoionised gas reprocesses the ionizing radiation into ultraviolet,
optical and infrared continuum and line emission. Variations in the central source launch spherical waves of
heatingandionizationthatexpandatthe speedoflightthroughthe surroundinggas. Eachchangeinionization
triggers a corresponding change in the reprocessed emission. In AGNs, the reprocessing time (hours) is small
compared with light travel time (days), so that the delay seen by a distant observer is dominated by the light
travel time.
The iso-delay surfaces are ellipsoids with one focus at the central source and the other at the observer, and
arewellapproximatedbyparaboloidsinthe reprocessingregionnearthe centralsource(Fig. 1). The time delay
is
R
τ = (1+cosθ) , (1)
c
for a reprocessing site at a radius R from the nucleus and azimuth θ, measured from 0 on the far side of the
nucleus to 180◦ on the line of sight between the nucleus and the observer. The delay is 0 for gas on the line of
sight between us and the nucleus, and 2R/c for gas directly behind the nucleus (Fig. 1). The time delay map,
Ψ(τ), is a 1-dimensional map of the emission line region,effectively slicing up the region along the nested set of
iso-delay paraboloids.
1.3. The Driving Lightcurve
The ionizing radiation that drives reprocessing includes EUV photons that are strongly absorbed by neutral
hydrogen in the interstellar medium, and are therefore not directly observable. Fortunately, we see nearly co-
temporal variations in continuum lightcurves at most ultraviolet and optical wavelengths, suggesting that the
continuum forms in a region much smaller than the emission line regions. The continuum light curve f (t) thus
c
serves as a useful surrogate for the unobservable light curve of the ionizing radiation.
1.4. Linear and Linerized Reverberation Models
Inthesimplestlinearreverberationmodel,reprocessingsitesspanarangeoftimedelays,andthelinelightcurve
f (t) is a weighted sum of time-delayed copies of the continuum light curve f (t),
ℓ c
∞
f (t)= Ψ (τ) f (t τ) dτ. (2)
ℓ Z ℓ c −
0
In this convolution integral, Ψ (τ) is the “transfer function”, or “convolution kernel”, or “delay map” of the
ℓ
emission line ℓ. This describes the strength of the reprocessed emission that arises from the regions between
pairs of iso-delay paraboloids. Of course each emission line has its own time variations f (t) and corresponding
ℓ
delay map Ψ (τ).
ℓ
Echo mapping aims to recover Ψ (τ) from measurements of f (t) and f (t) made at specific times t . To fit
ℓ ℓ c i
suchobservations,the model aboveis too simple in severalrespects. First, additionalsourcesof lightcontribute
to the observed continuum and emission-line fluxes. Examples are background starlight, and narrow emission
lines. Since these sources do not vary on the reverberation timescale, they simply add constants to f (t) and
ℓ
f (t),
c
∞
f (t)=fB+ Ψ (τ) f (t τ) fB dτ. (3)
ℓ ℓ Z ℓ c − − c
0 (cid:2) (cid:3)
Thislinearizedmodelalsousefulasatangentlineapproximationtoanon-linearlineresponse. The“background”
fluxes,fB forthelineandfB forthecontinuum,aresetsomewherenearthemiddleoftherangeofvaluesspanned
ℓ c
bytheobservations. ThedelaymapΨ (τ)thengivesthe marginalresponseofthelineemissionfromgasattime
ℓ
delay τ, to changes in ionising radiation above or below the chosen backgroundlevel.
1.5. Inversion Methods
Three practical methods have been developed for deriving Ψ (τ) from observations. The Regularized Linear
ℓ
Inversion(RLI) method2,3 and the Subtractive Optimally-Localized Averages (SOLA) method,4 use the linear
reprocessing model to directly invert well-sampled equally-spaced or interpolated lightcurves. The Maximum
EntropyMethod(MEM),isusefulforthemoresophisticatedfittingproblems,allowingfornon-uniformsampling
and data quality in the lightcurves, and non-linear reprocessing models. The echo mapping results presented
in this paper are obtained with the maximum entropy echo mapping code MEMECHO,5,6 which makes use of the
maximum entropy fitting code MEMSYS.7
2. FIRST-GENERATION ECHO MAPPING EXPERIMENTS
2.1. Size and Radial Structure
To illustrate the quality of the echo maps constructed from current datasets, Fig. 2 shows a MEMECHO fit of the
linearized echo model of Eqn. (3) to Hβ and optical continuum lightcurves of NGC 5548.5 The lightcurves
are extracted from optical spectra arising from the 9-month AGN Watch campaign in 1989. Data from many
observatoriesare inter-calibrated,and the errorbars areestimated from the internal consistency of independent
measurements made close together in time. Subtracting the continuum background fB, convolving with the
c
delay map Ψ (τ), and adding the line background fB, gives the Hβ light curve. The three fits shown, all with
ℓ ℓ
χ2/N =1, give some impression of the uncertainty in the fit arising from the noise level of the data and gaps in
time coverage.
Figure 2. EchomapsofHβ emission inNGC5548recoveredbyfittingtodatafrom a9-monthAGNWatchmonitoring
campaign in1989. Theopticalcontinuumlightcurve(lowerpanel) isconvolvedwiththedelaymap(topleft) toproduce
theHβ emission-linelightcurve(topright). Horizontaldashedlinesgivethemeanlineandcontinuumfluxes. Threefits
are shown to indicate likely uncertainties. Data from (Ref.5).
The delay map of Hβ emission in NGC 5548 rises to a peak at 20 days, and declines to low values by 40
days. This measures the size of the Hβ emission-line region, 20 light days. Tests with simulated data using
∼
the same time sampling and signal-to-noise ratios indicates that the resolution achieved in this map is about
10 days. Similar maps for a variety of ultraviolet emission lines indicate that high ionisation lines have smaller
delays than low-ionisation lines.8 This implies that reprocessing occurs over a wide range of radii, with higher
ionisation closer to the nucleus.
2.2. Velocity-Delay Maps
To derive a Doppler-delay map from observations, simply slice the observed spectra into wavelength bins, and
recover a delay map from the light curve at each wavelength. This is a simple extension of the echo model used
to fit continuum and emission-line lightcurves. At each wavelength λ and time t, obtain the emission-line flux
∞
f (v,t)=fB(v)+ f (t τ) fB Ψ (v,τ) dτ (4)
ℓ ℓ Z c − − c ℓ
0 (cid:2) (cid:3)
by adding time-delayed copies of the continuum light curve to a time-independent backgroundspectrum fB(v).
ℓ
In Fig. 3, the velocity-delay map for Civ 1550 and Heii 1640 emission was reconstructed from 44 IUE
spectra of NGC 4151.10 While spanning only 36 days, this campaign recorded favorable continuum variations,
including a bumpy exponential decline followed by a rapid rise, which were sufficient to support echo mapping
on delays from 0 to 20 days. Although a strong Civ absorption feature obliterates the delay structure at small
velocities, it is clear that a wide range of velocities is present at small delays, and and smaller range at larger
delays. The dashed curves in Fig. 3 give escape velocity envelopes v = 2GM/cτ for masses 0.5, 1.0, and
2.0 107 M . This clearly supports the hypothesis of virial motions, and spuggests a mass of order 107 M .
⊙ ⊙
×
2.3. Black Hole Masses
With the assumption of virial motions, black hole mass can be estimated from
c τ σ2
M =f , (5)
G
Figure 3. Velocity-delaymapofCiv 1550emission(andblendedHeii 1640emission)inNGC4151recoveredbyfitting
to 44 IUE spectra. Absorption wipes out the core of the Civ line. Dashed curves give escape velocity envelopes for
masses 0.5, 1.0, and2.0×107M⊙. Aninversecorrelation between velocity width and timedelay supportsthehypothesis
of virial motions, and yields virial mass estimates for dozens of supermassive black holes. Black hole masses estimated
from the Hβ emission line, based on cross-correlation time delay and FWHM of the line in the variable component of
thelight, are from (Ref.9).
where τ and σ are the time delay and velocity dispersion of an emission line, and f is a form factor allowing
for uncertain details of the flow geometry, kinematics and orientation. The virial hypothesis predicts that lines
formed at different radii in the same object should obey σ τ−1/2. The internal consistency of mass estimates
∝
fromdifferentlinesinNGC5548,8,11 andinafewotherobjects,12 supportsthevirialhypothesis. Anuncertainty
of perhaps a factor of 3 remains due to the factor f representing ambiguity in the detailed kinematics of the
flow.13
Reverberation masses ranging from 106 to 109M are now available for several dozen AGNs9,14 based on
⊙
cross-correlation time delays and rms widths of Hβ emission (Fig. 3). At present there are no reverberation
masses for high-redshift AGNs. However, the low-redshift reverberation masses serve to calibrate an empirical
mass-radius-luminosityrelationshipfromwhichblackholemassesmaybeestimatedformoredistantsystems.9,14
This extends black hole mass estimates to all AGN with a measured Hβ line width. Such estimates are useful
in the investigation of the evolution of black hole masses and accretion rates.
2.4. Temperature Profiles of AGN Accretion Disks
In theory, steady-state accretion disks have a T (MM˙ )1/4R−3/4 structure, and give rise to a characteristic
spectrum f (MM˙ )2/3λ−1/3D−2. Here M˙ is∝the accretion rate and D is the distance. AGN spectra are
ν
∝
generally much redder than this, casting doubt on the validity of the disk model. We can measure the T(R)
profilesofAGNaccretiondisksbymeasuringλ-dependentcontinuumreverberationsarisingfromthedisksurface.
With T(R) decreasing outwards, reverberations at smaller R and shorter λ will preceed those at larger R and
longerλ. Timedelaysincreaseasτ R/c,andblackbodyspectrapeakatT hc/kλ,sothatshorterwavelengths
≈ ≈
sense disk annuli at higher temperatures. A disk surface with T R−b will reverberates with a delay spectrum
τ λ−1/b. The T (MM˙ )1/4R−3/4 profile of a steady-state disk∝corresponds to τ (MM˙ )1/3λ4/3.
∝ ∝ ∝
AGN continuum lightcurves at different wavelengths exhibit a high degree of correlation, with time delays
generally shorter than a few days. In the best-observed case,15 NGC 7489, the time delay does increase with
wavelength,consistentwithτ λ4/3. Moreover,subtractingbrightandfaintspectrayieldsadifferencespectrum
∝
closetof λ−1/3. Theseresultssuggestthatvariabilitymaybe aneffective wayto isolatethe diskcomponent
ν
of AGN sp∝ectra. The NGC 7489 results yield estimates for MM˙ and D, leading to an encouragingly sensible
estimate of the Hubble constant, H √cosi 42 9km s−1/Mpc. If similar results are obtained for a larger
0
≈ ±
sample of AGNs, spanning a range of redshifts, the results may be used to to probe cosmology beyond the
redshift horizon of supernovae.
2.5. Highlights of First-Generation Echo Mapping Experiments
The first decade of echo mapping experiments, a series of intensive monitoring campaigns undertaken in par-
ticular by the AGN Watch consortium(http://www.astronomy.ohio-state.edu/~agnwatch/),has sharpened
our knowledge of AGNs in numerous ways. The basic photoionisation picture is well supported by the corre-
lated emission line and continuum variations, with lines lagging behind the continuum, in several dozen nearby
AGNs. Interpreting the time delay, τ, as light travel time, the inferred BLR sizes, R cτ, are 10-100 times
∼
smallerthanexpectedfromearliersingle-cloudphotoionizationmodels. Thesmallersizeimplieshigherdensities,
n 1011cm−3. Higher-ionization lines have smaller delays and larger widths,16 suggesting a radially-stratified
∼
ionization structure with virial kinematics.11 Larger time delays in more luminous sources imply R L0.5−0.7,
∝
compatible with an ionization-bounded BLR.14 In several sources, an anti-correlation between velocity disper-
sionsandtime delaysis consistentwith virialmotions.8,11,12 Onthis basis,“reverberationmasses”areavailable
for black holes in AGNs for which the Hβ velocity dispersion σ and time delays τ are measured. This in turn
calibrates an empirical M–σ–L relationship from which black hole masses can be estimated for all AGNs with
measured Hβ line widths.9 We thus have a foundation for demographic studies of the growth of supermassive
black holes over cosmic time.
3. NEXT-GENERATION ECHO MAPPING EXPERIMENTS
In the next decade, striking improvements in echo mapping capabilities are expected from technology develop-
ments on two fronts.
First, improvements in the quality and time sampling of spectrophotometric monitoring datasets should
dramatically improve the resolution and fidelity of the maps. The present 5-10 day resolution of echo maps is
limitedbythenoiselevels(typically3%),cadence,anddurationofthelightcurves. JustastheVLAdramatically
improved the U V coverage and hence resolution and fidelity of radio maps, so the echo maps will be greatly
−
improved by using facilities that are specifically designed for and dedicated to sustained spectrophotometric
monitoring.
RoboNet(http://star-www.st-and.ac.uk/~kdh1/jifpage.html)isaglobalnetworkof2-mrobotictele-
•
scopes equipped with identical multi-band CCD imagers and integral-field unit spectrographs.
Queue-scheduled spectrographs on large ground-based telescopes are ideal for sustained daily monitoring,
•
and will enable echo mapping of fainter AGNs at larger redshifts.
Kronos (http://www.astronomy.ohio-state.edu/~kronos) is a small space telescope (NASA MidEx)
•
equippedwithsimultaneousX-ray,UV,andopticalspectrographsina14dorbittoenablenearlycontinuous
coverage for weeks to months and daily sampling for hundreds of days.
What may we expect to achieve in the next-generation echo mapping experiments? Simulation tests, some of
which are presented below, indicate that high fidelity echo maps with a resolution exceeding 1 light day will
emerge from datasets with a time sampling <1 day, a duration 300 days, and accuracies of 1%.
∼ ∼
Second, in tandem with better data, better models are needed to interpret the improved datasets. We cur-
rently extract lightcurves from observed spectra, and fit those lightcurves with a simple linear or linearised
reprocessing model to construct delay maps for each line or wavelength independently of the others. This rel-
atively model-independent approach ignores a great deal of prior information. Photoionisation models predict
highlynon-linearandanisotropicresponsesthataredifferentforeachemissionline. Giventheevidencesupport-
ing the photoionisation hypothesis, we should now aim to improve the echo maps by building in the additional
constraints from photoionisation physics. The next-generation reverberation models, currently under develop-
ment, will fit the entire reverberatingspectrum, avoidingthe needto extractlightcurvesand de-blendlines, and
build photoionisation physics into the fitting process. This approach should yield 5-dimensional maps of the
geometry (R, θ), kinematics (V), and physical conditions (n,N) in the photoionised gas.
We discuss and illustrate these developments below.
Figure4.Twosimulations,eachillustratingtherecoveryofadelaymap(topleft)fromdatapointssamplinganerratically
varying continuum light curve (bottom) and the corresponding delay-smeared emission-line light curve (top right). The
data points sample 400 epochs spaced by 1 day. In simulation 1 (left) the true map is Ψ (τ)∝τe−τ/10. In simulation 2
ℓ
(right)themaphas5sharppeaksatdelays5,10,20,40,and80days. Inbothcasesthereconstructeddelaymapsclosely
resemble thetrue maps.
3.1. Better Datasets, Sharper Maps
Fig. 4 illustrates the recovery of 1-dimensional delay maps based on MEMECHO fits to simulated lightcurves. The
time sampling and signal-to-noise ratios in these simulations are designed to resemble datasets that will arise
routinely with next-generation spectrophotometric monitoring facilities, such as Kronos and RoboNet.
In both simulations, examine and compare the driving continuum lightcurve (bottom) with the responding
line lightcurve (top right). The delayed maxima and minima indicate that the delay map has a mean delay
of about 20 days. The fast variations evident in the continuum lightcurves appear to be washed out in the
line lightcurves, which are much smoother, indicating that a range of delays is present. By eye, this is about
all that we can safely infer. The line lightcurve in simulation 2 does exhibit more fast structure than that in
simulation 1, but this structure is not cleanly correlated with the fast continuum variations. It is not obvious
what this may imply about the delay structure. Evidently our eyes and brains have a rather limited ability to
interpret reverberationdatasets by inspection – much as is also the case for U V visibility measurements that
−
are employed in interferometry.
When MEMECHO fits the two lightcurves in detail, however, information from all the small but significant
changes recorded in the lightcurves is assembled to construct the delay map (top left). The two maps recovered
byMEMECHOarequitedistinct. Insimulation1thedelaymaphasasmoothdistribution,risingtoaroundedpeak
and then declining smoothly. In simulation 2 the delay map has 5 discrete peaks. In both cases the recovered
maps closely resemble the true map. With such high quality data, the delay map is recoveredwith high fidelity.
There is some blurring, of course, because no finite dataset can recover the map with infinite resolution. The
resolution achieved is about 1 light day.
Fig.5examinesMEMECHOreconstructionsof2-dimensionalvelocity-delaymapsfromsimulatedKronosdatasets.
This simulation illustrates the extent to which fine-scale structure in the geometry and kinematics of an AGN
emission line region can be recovered from a Kronos dataset. The simulated dataset, designed to resemble a
KronosobservationofNGC5548,includesrealisticnoiselevelsallowingfortheeffectiveareacurvesoftheKronos
spectrographs,and assuming that a 1 hour spectrum is taken every 4.8 hours for a total of 200 days. We adopt
thepower-lawmodelofKaspi&Netzer,17 inwhichthedensitynandcolumndensityN arepower-lawfunctions
of radius, and the photoionisation code ION is used to evaluate the anisotropic and non-linear responses of the
emission lines at each level of ionising continuum flux. For the geometry and kinematics we adopt a Keplerian
Figure 5. High-fidelity velocity-delay maps of Lyα and Civ emission reconstructed from a simulated Kronos dataset
spanning 200 days. The emission is assumed to arise from a Keplerian accretion disk with a two-armed trailing spiral
density wave pattern (left). The spiral waves are clearly visible in the true velocity-delay map (middle) and in the map
recoveredfromthesimulatedKronosspectra(right). Dashedcurvesgivethemaximumtimedelayallowedateachvelocity
for bound gas around a black hole of mass 3×107M⊙.
disk with a 2-armed trailing spiral density wave. The disk is comprised of a large number of discrete clouds
orbitingablackholeofmass3 107M . The co-planarKeplerianorbitsareelliptical, sothateachcloudspends
⊙
×
more time near apbothron than peribothron, and are rotated by an angle proportionalto the logarithmof their
semi-major axes, thereby producing the spiral density wave. From the synthesized spectra, we fit continua,
extract line lightcurves at each velocity, and use MEMECHO to construct velocity-delay maps for each line.
In Fig. 5, the spiral waves clearly visible on the sky view (left panel) extend out to 10−4 arcsec (the
≈
redshift of NGC 5548 is z = 0.016). This structure is beyond the reach of foreseeable developments in direct
or interferometric imaging technology. Yet the velocity-delay map (middle panel) clearly shows this spiral wave
structure,anditis alsoclearlyrecognizableinthe reconstructedvelocity-delaymap(rightpanel)recoveredfrom
the simulated Kronos dataset.
These results are typical ofmany other simulations that we haverun, for a wide variety of source structures,
indicating that major advances in the fidelity and information content of echo maps will arise when the data
quality is significantly improved. The simulations indicate that we can sharpen our vision by a factor of 10
compared to the 5-10 day resolution that has been achieved to date.
3.2. 5-Dimensional Maps: Ψ(R,θ,n,N,v)
Inparallelwithimprovementsinthereverberationdatasets,wemustimprovethemodelsthatweusetointerpret
thedata. Agreatdealofinformationonthegeometry,kinematics,andphysicalconditionsofblackholeaccretion
flows is encoded in the detailed reverberations of emission-line spectra. We cannot extract this information by
inspection of the data. To tap this information, and reconstruct maps of black hole environments, we must aim
to fit the observedreverberationsin far greaterdetail than has previously been attempted. The next-generation
echo-mappingcodes, building in constraintsfrom photoionisationphysics, are currentlyunder development. We
sketch the new methods, and then illustrate some preliminary results below.
Rather than extracting line and continuum lightcurves, we now synthesize and fit f (λ,t), the complete
ν
spectrum at each time. This avoids sticky problems of continuum fitting and de-blending the overlapping wings
ofemissionlines. Wemodelthespectrumasasumofthreecomponents: directlightfromthenucleus,reprocessed
light from the surrounding gas clouds, and backgroundlight:
f (λ,t)=fD(λ,t)+fR(λ,t)+fB(λ) . (6)
ν ν ν ν
The background light, fB(λ), allows for contamination by non-variable sources, e.g. starlight from the host
ν
galaxy. The direct light from the nucleus is
L(t)S (λ)
fD(λ,t)= ν , (7)
ν 4πD2
whereDisthedistance,L(t)isthelightcurve,andS (λ)istheshapeofthespectrumemergingfromthenucleus.
ν
The emission-line spectrum of the reprocessed light is
∞ L(t τ)
fR(λ,t)= − Ψ (λ,τ,L(t τ)) dτ , (8)
ν Z 4πD2 ν −
0
arising from a non-linear transfer function given by
Ψ (λ,τ,L) = 2πR dR sinθ dθ dn dN dv
ν ℓ
ΨP(R,Rθ,RnR,NR,v) ǫℓ(F,n,N,θ) (9)
×
g (λ (1+v/c)λ ) δ R L 1/2 δ τ R(1+cosθ) .
× ν − ℓ (cid:16) −(cid:0)4πF(cid:1) (cid:17) (cid:0) − c (cid:1)
This at first sight frighteningly complicated expression is simply a sum of contributions over all the emission
lines, ℓ,integratedoverthe geometricvolume,2πR dR sinθ dθ, andthe differenttypes of gasclouds,dn dN dv.
The5-dimensionalmapdescribingthepopulationofgascloudsisthedifferentialcoveringfactor,Ψ(R,θ,n,N,v).
Each type of cloud is specified by 5 parameters, the density n, column density N, distance from the nucleus R,
azimuth θ, and Doppler shift v. ∗ At the observers’stime t, a cloud at radius R and azimuth θ is exposed to an
ionizing photon flux
L(t τ)
F(t,R,θ)= − , (10)
4πR2
which allows for the light travel time delay
R
τ = (1+cosθ) . (11)
c
The irradiated cloud responds by emitting line ℓ with an efficiency ǫ (F,n,N,θ), which we evaluate using a
ℓ
photoionisation code, e.g. CLOUDY. Each line has a different reprocessing efficiency, depending on the incident
ionisingfluxF,the clouddensityn andcolumnN ,theviewingangleθ †,andelementabundances. Finally,the
Gaussianfrequencydistributiong appliesthe Dopplershift, whereλ is the restwavelengthofthe line,andthe
ν ℓ
two Dirac δ distributions ensure that the correcttime delay τ and ionizing flux F are used at each reprocessing
site.
To fit the above model to observations of f (λ,t), we must adjust the distance D, the ionizing radiation
ν
lightcurve L(t), the backgroundspectrum fB(λ), andthe 5-D cloudmap Ψ(R,θ,n,N,v). Constraintsavailable
ν
from reverberating emission-line spectra, perhaps 104 data points, are insufficient to uniquely determine the
∼
5-dimensional cloud map, Ψ, which may have 106 pixels. Our computer code MEMBLR uses the maximum
∼
entropy method to locate the “simplest” cloud maps that fit the data. Desktop computers are now fast enough
to support this type of detailed modelling and mapping of AGN emission regions. Note that because every line
provides a different weighted average, a different projection, of the 5-dimensional cloud distribution, we have a
generalised form of tomography.
3.3. Recovery of a Hollow Shell Geometry
In Fig. 6 we present results of a simulation test designed to investigate how well the geometry and physical
conditions may be recoverable from reverberations in ultraviolet emission-lines. In this simulation, we assume
∗Theclouds also haveaposition angle φ around thelineof sight, and twoperpendicularvelocity components,butwe
omit them here because thedata are unchangedif we rotate the cloud distribution around the line of sight.
†The viewing angle θ is thesame as theazimuth.
Figure 6. Reconstructed map Ψ(R,θ,n) of a thin spherical shell (right) from a maximum entropy fit to 7 ultraviolet
emission-linelightcurves(left). Thelightcurvefitsachieveχ2/N =1. TheshellradiusR/c=12danddensitylogn=11
are correctly recovered. For furtherdetails, see text.
thatcloudswithdensityn=1011cm−3 andcolumnN =1023cm−2areuniformlydistributedoverathinspherical
shell of radius R/c = 12d. An erratically varying source at the centre of the hollow shell drives reverberations
in 7 ultraviolet emission lines. We compute the emission-line lightcurves, allowing for light travel time delays
and using CLOUDY to account for each line’s anisotropic and non-linear reprocessing efficiency. We sample the
synthetic lightcurvesat121epochsspacedby 2days,andaddnoise to simulateobservationalerrors. Finally,we
reconstruct a 3-dimensional cloud map Ψ(R,θ,n) by fitting to the 7 synthetic lightcurves.
It is important to realise that this MEMBLR fit does not assume a spherical shell geometry, but rather it
considers every possible cloud map, Ψ(R,θ,n), and tries to find the “simplest” map that fits the emission-line
lightcurves. The fit adjusts 4147 pixels in the cloud map Ψ(R,θ,n), 143 points in the continuum light curve,
f (t), 7 emission-line background fluxes, fB(λ), and 1 continuum flux, fB. This fit assumes the correct column
c ℓ c
density,N, anddistance,D. The fit toN =968datapoints achievesχ2/N =1. Entropymaximization“steers”
each pixel in the map toward its nearest neighbors, thus giving preference to “smooth” maps, and toward the
pixel with the opposite sign of cosθ, thus giving preference to maps with front-back symmetry.
On the left-hand side of Fig. 6, we see that the 8 lightcurves, 7 lines and 1 continuum, are well reproduced
by the fit. The highs and lows are a bit more extreme in the data – a common characteristic of regularized
fits. To the left of each emission-line light curve, three delay maps are shown corresponding to the maximum
brightness (solid curve), minimum brightness (dashed), and the difference (dotted). All the lines exhibit an
inward anisotropy except Ciii] . All the lines have positive linear responses except Mgii , which has a positive
response on the near side of the shell and a negative response on the far side.
Onthe right-handsideofFig.6,weseethatthe fittothe 8lightcurvesrecoversahollowshellgeometrywith
the correct radius. The density-radius projection of the map has a peak at the correct radius and density. The
